Analysis of Static Contact Problems

Abstract

In this chapter we illustrate the use of the abstract results obtained in Chap. 4, in the study of three representative static frictional contact problems for deformable bodies. In the first two problems we model the material’s behavior with a nonlinear elastic constitutive law and with a viscoelastic constitutive law with long memory, respectively, and we describe the frictional contact with subdifferential boundary conditions. In the third problem the deformable body is assumed to be piezoelectric and, therefore, we model its behavior with an electro-elastic constitutive law. And, again, the contact conditions, including the electrical conditions on the contact surface, are of subdifferential type. For each problem we provide a variational formulation. For the first two problems it is in a form of a hemivariational inequality for the displacement field and, for the third one, it is in a form of a system of hemivariational inequalities in which the unknowns are the displacement and electric potential fields. Then, we use the abstract existence and uniqueness results presented in Chap. 4 to prove the weak solvability of the corresponding contact problems and, under additional assumptions, their unique weak solvability. Finally, we present concrete examples of constitutive laws and frictional contact conditions for which our results work and provide the related mechanical interpretation. Everywhere in this chapter we use the notation introduced in Chap. 6.